Complex-Momentum, Spectral, and Real-Time Information

Euclidean functional equations determine correlation functions on Euclidean momenta. Poles, cuts, widths, and real-time response live in a different analytic domain. Exact Euclidean data satisfying the full field-theory axioms determine the corresponding Lorentzian theory, but a finite, noisy, truncated dataset does not determine a unique spectral density without extra analytic assumptions. Direct complex-momentum solution, a spectral ansatz, and numerical reconstruction are distinct methods with distinct failure modes.

Required background. Schwinger–Dyson hierarchies supplies the Euclidean equations, while spectral decomposition supplies poles and continuum support. Helpful background. Euclidean correlators and Schwinger functions supplies reflection and reconstruction conditions.

Spectral representation and its assumptions

For a zero-temperature bosonic operator with a positive spectral measure,

$$G_E(p_4,\mathbf p) =\int_0^\infty\mathrm d\omega\, \frac{2\omega\,\rho(\omega,\mathbf p)} {\omega^2+p_4^2}, \qquad \rho(\omega,\mathbf p)\ge0.$$

Subtractions can be required for ultraviolet growth. A stable one-particle state contributes a delta function; multiparticle states produce continuum support. The retarded function follows from the boundary value

$$G_R(\omega,\mathbf p) =-G_E\!\left( p_4=-i(\omega+i0^+),\mathbf p \right)$$

in the stated convention.

Positivity is operator-dependent. It applies to physical positive-metric operators under the Osterwalder–Schrader assumptions, not automatically to gauge-fixed gluon, ghost, or quark propagators. A sign-changing reconstructed gauge-fixed spectral function is therefore not by itself an inconsistency or a gauge-invariant confinement proof. The exact Euclidean reconstruction theorem and its positivity hypothesis are given in Osterwalder and Schrader 1973, pp. 86–94.

The Källén–Lehmann representation and its pole-plus-continuum interpretation are developed in Lehmann 1954, pp. 342–357.

Why Euclidean inversion is ill posed

At fixed $\mathbf p$, the map

$$\rho(\omega) \longmapsto G_E(p_4)$$

is a smoothing integral transform. After discretization,

$$G_i =\sum_jK_{ij}\rho_j.$$

The singular values of $K$ decay rapidly. Components of $\rho$ along small-singular-value directions can change greatly while changing $G_i$ less than its uncertainty. Therefore:

• a zero solver residual for the Euclidean equation does not imply a resolved spectrum;
• many pole-and-continuum shapes can fit the same finite data;
• regularization, entropy, smoothness, positivity, or a basis choice supplies information not present in the data;
• error bars require the full Euclidean covariance, not pointwise errors alone.

Jarrell and Gubernatis analyze this inverse-problem structure for spectral reconstruction Jarrell and Gubernatis 1996, §§ II–IV.

Three routes to real-time information

Direct complex-momentum solution

Solve the functional equations for complex external momenta and deform loop contours consistently around propagator and kernel singularities. This can expose poles and cuts without an intermediate inversion, but it requires:

• a known analytic domain for every input function;
• contour deformations that include crossed residues when required;
• branch conventions and Riemann sheets;
• checks that regulator singularities do not enter the physical domain.

A Euclidean solution interpolated by a convenient entire function is not a direct complex solution unless the interpolation’s analytic structure is justified.

Spectral representation inside the equations

Insert spectral representations for internal propagators and perform frequency integrals analytically or semi-analytically. This preserves causal structure when the representation is valid. It can fail when the field lacks a positive spectral density, when complex-conjugate singularities are required, or when a finite spectral ansatz excludes the true cut structure.

Reconstruction from Euclidean data

Infer $\rho$ with a prior or regularizer. The output is conditional on:

• positivity or sign freedom;
• support and asymptotic assumptions;
• number and type of pole components;
• regularization strength or prior covariance;
• data window and covariance.

Features stable only under one prior are prior-dependent. A defensible result publishes resolution tests with synthetic spectra, held-out Euclidean points, and basis or prior variation.

Poles, cuts, and sheets

A stable particle corresponds to a first-sheet pole on the real mass shell. A resonance corresponds to a pole on an analytically continued sheet and manifests through a cut on the physical sheet. A Euclidean fit of

$$G_E(p^2)\approx\frac{Z}{p^2+M^2}$$

over a finite spacelike interval does not distinguish a true pole from a narrow continuum approximation. One must establish the continuation domain or use an observable sensitive to the discontinuity.

For bound-state equations, the same issue appears when $\lambda(P^2)=1$ is extrapolated from $P^2>0$ to $P^2=-M^2$. The extrapolation error belongs to analytic continuation, not to the eigenvalue solver.

Validation battery

A spectral claim should report separately:

1. Euclidean equation and data residuals;
2. truncation and regulator variation of the Euclidean input;
3. continuation or reconstruction assumptions;
4. mock-data resolution and false-positive tests;
5. sum rules, asymptotic moments, and reality constraints;
6. positivity tests only when the operator warrants them;
7. comparison with an independent real-time, scattering, or finite-volume observable.

The functional-equation closure and validation map places analytic assumptions after branch selection. The functional-method validation comparison requires continuation variation, data covariance, and honest reporting of unresolved features.

Common pitfalls

Invoking uniqueness from exact reconstruction theorems for finite data. The exact theorems assume complete correlators and axioms. A finite numerical array does not meet those hypotheses.

Interpreting a Euclidean fit pole literally. Several analytic functions can agree within errors on the spacelike interval and differ in their singularities.

Demanding positivity of a gauge-fixed field. Reflection positivity is a statement about the physical positive-metric reconstruction. Gauge-fixed elementary fields can violate it.

Exercises

1. Let $K=U\operatorname{diag}(s_i)V^{\mathsf T}$ with $s_i\to0$. Show why a perturbation along $V_i$ is poorly constrained.

Solution

Changing $\rho$ by $\delta\rho=cV_i$ changes the data by

$$\delta G=c\,s_iU_i.$$

For small $s_i$, $c$ can be large while $\lVert\delta G\rVert$ stays below the data uncertainty. Regularization suppresses such components by assumption; it does not make them measured.

2. A reconstruction finds a narrow peak that disappears when the smoothness prior is weakened. How should it be reported?

Solution

It is not a data-resolved pole. Report it as a prior-dependent feature or an upper bound on resolvable width, together with mock-data resolution tests. A physical-pole claim requires stability under justified priors or an independent analytic or scattering check.

Continue

Functional-Method Validation and Error Control combines continuation uncertainty with truncation, branch, and numerical errors. Gauge Fixing, BRST Constraints, and the Gribov Problem explains why positivity has a different status for gauge-fixed fields.

References

• Jarrell, Mark, and J. E. Gubernatis. “Bayesian Inference and the Analytic Continuation of Imaginary-Time Quantum Monte Carlo Data.” Physics Reports 269 (1996): 133–195. DOI.
• Lehmann, Harry. “On the Properties of Propagation Functions and Renormalization Constants of Quantized Fields.” Il Nuovo Cimento 11 (1954): 342–357. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI.